Long run Cost Functions for Electricity Transmission

Transcription

1 Long run Cost Functions for Electricity Transmission JUAN ROSELLÓN Centro de Investigación y Docencia Económicas (CIDE) and Dresden University of Technology (TU Dresden) INGO VOGELSANG Department of Economics, Boston University HANNES WEIGT Dresden University of Technology (TU Dresden) Conference on The Economics of Energy Markets, IDEI Toulouse, January 28 and 29,

2 Contents Introduction Characterization of Electricity Transmission Outputs Model, Topologies and Data Scenarios and Results Conclusion 2

3 Introduction 3

4 Electricity transmission differs from other transportation and network systems (pipelines, railroads, the road system, telecommunications). The physical laws of electricity make transmission complex and unusual. No characterization of cost function for transmission grids. Two reasons: First is that there is no full agreement on an obvious output. Second is that the effects of Kirchhoff s laws lead to bewildering irregularities in the relationship between outputs and capacities. Transmission cost functions: strange properties that make them interesting for an audience outside electricity. Where else can you expect to have negative marginal costs? Knowledge of transmission cost functions help to plan cost minimizing transmission systems over a wide range of potential outputs. It could also help to assess investments in renewable energies, and in the implementation of incentive mechanisms for transmission investment. 4

5 We study long run electricity transmission cost functions based upon a definition of transmission output in terms of point to point transactions Build on Hogan, Rosellón and Vogelsang (HRV) (2010) model (which combines the merchant and regulatory approaches): FTR based cost functions exhibit very normal economic properties in a variety of circumstances if topology is given, and when capacity of lines can be changed. One purpose of our study is to establish that the problem of non wellbehaved, non continuous transmission cost functions is related to demand changes that lead to a change in network topology (as suggested by HRV, 2010). We test the behavior of FTR based cost functions for distinct network topologies. Two basic cases: First adjust line capacities, but nodes, lines, impedances and thus PTDFs do not change. Second case allows for changes in line impedances (and PTDFs) correlated to the changes of line capacities. 5

6 Characterization of Electricity Transmission Outputs 6

7 The definition of the output for electricity transmission is difficult since the physical flow through a meshed transmission network is complex and highly interdependent among transactions (Bushnell and Stoft,1997, and Hogan, 2002a, 2002b). Under a network with loop flows, outputs could be defined as bilateral trades between pairs of nodes that aggregate to net injections at all nodes. This idea derives from the FTR literature. An FTR (obliglation) q ij represents the right to inject electricity in the amount of q at node i and to take delivery of the same amount at node j. The FTR does not specify the path taken between i and j. It is a flow concept. Whereas in directed networks like natural gas or oil an additional unit of output can normally be associated with a well defined cost parameter or function, additional output in electricity networks depends on the grid conditions. 7

8 Model, Topologies and Data 8

9 The network topology (nodes and lines) is described by the network incidence matrix (Léautier, 2000). For a given network topology we assume that the line capacity is variable so that it can be changed between 0 and, but at a cost. Transmission cost function c(.): least costs combination of line capacities k necessary to satisfy Q ij (the matrix consisting of a specific set of FTR combinations q ij ): Minimization is subject to technical restrictions representing the network s power flow characteristics: line capacity constraint energy balance constraint 9

10 We incorporate model in General Algebraic Modeling System (GAMS) as a non linear minimization tool. Figure 1: Two network topologies 10

11 Four forms of line extension costs functions f ij (k ij ): constant marginal cost, decreasing marginal cost (economies of scale), increasing marginal costs (diseconomies of scale), and lumpy behavior. Linear function (constant marginal cost): Logarithmic function (economies of scale): Quadratic function (diseconomies of scale): Lumpy function: 11

12 Scenarios and Results 12

13 Fixed line reactances: extending one FTR 13

14 Fixed line reactances: global cost function 14

15 Variable line reactances: extending one FTR 15

16 Variable line reactances: global cost function 16

17 17

18 Conclusions regarding the relationship of kinks, negative slopes and loop flows for the non lumpy cost functions. First, smoothness is gained with variable reactances in the three node case and in the global cost function case: Whereas in the fixed reactance scenario it was clear that for specific FTR combinations the power flow on one line will fall to zero and cause kinks in both the 1 FTR and global cost function cases, this is not valid in the variable reactance scenario because the canceling out points can change with the alteration of network characteristics. Additionally, for the global cost function case the counter flow structure may imply decreasing ranges of the cost function. Lumpy extension case: gains from considering variable reactances will imply an increasing stepwise function. Results then indicate that taking into account the full characteristics of electricity networks provides a cost framework that is closer to the well behaved continuous functionality. 18

19 Comparative analysis suggests piece wise continuity of cost functions. This property crucial is for the application of price cap incentive mechanisms to real world expansion projects. Redefinition via a FTR approach only takes into account the overall extension costs, and thus avoids line specific discontinuities. More realistic scenario when cost minimization occurs over the optimal design of the network (location and number of links and nodes denoted by transfer admittance matrix H) H becomes a variable, and a more complicated cost minimization problem results. The problem also leads to new goods (FTRs) for new nodes, and new goods change the costs for all of the old goods. The method would calculate the cost function for the changed network, and then compare it to the cost function of the original network. If the new cost function lies everywhere below (above) the original cost function, the new topology dominates (is dominated by) the original one. 19

20 Application of HRV (2010) Rosellón, J. and H. Weigt (2010), A Dynamic Incentive Mechanism for Transmission Expansion in Electricity Networks Theory, Modeling and Application, (winner of the Reimut Jochimsen Prize) Simplified model of the BENELUX: Implementation of the HRV model to meshed electricity networks. Covering 15 nodes and 28 lines. Including 8 plant types (nuclear, lignite, coal, CCGT, gas/oil, hydro, pump) with fixed marginal costs. Initial congestion between Belgium and France, and Germany and the Netherlands. Neglecting wind capacities. 20 periods, 8% interest rate. Only network upgrades possible at linear extension costs of 100 per km per MW capacity.

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22 Total transmission capacity is significantly increased, the Transco s profits are augmented, and there is convergence towards the welfare optimum. 22

23 Conclusions 23

24 Our simulations suggest that FTR based cost functions remain piecewise continuous as well as piecewise differentiable over the entire FTR range. Regions with negative marginal costs could cause concern. They seem to shrink as one moves to more realistic (multi node) networks and include changes in reactances. The introduction of a link between capacity and reactance appears to reduce the impact of loop flows in terms of significant kinks. Smoothness of non lumpy cost functions is gained with variable reactances. For modeling purposes, the logarithmic and lumpy behaviors produce high degrees of nonlinearities with non smoothness, and require further calculations and solver capabilities, the quadratic functions show a generally continuous behavior, and the linear extension functions fall somewhere between. Most suitable for modeling, therefore, is using the linear functions with the piecewise, linear nature of the resulting global costs function, which let us derive global optima. The ultimate research challenge is to identify the changes in network topology: number of configurations increases significantly with the number of nodes. We could further combine our analytical approach with actual (engineering) data to estimate actual cost functions in ways pioneered for telecommunications networks (Gasmi et al., 2002). 24

25 References Hogan, W., J. Rosellón and I. Vogelsang (2010), A Combined Merchant Regulatory Mechanism for Electricity Transmission Expansion, Revise and Resubmit, Journal of Regulatory Economics. Rosellón, J. and H. Weigt (2010), A Dynamic Incentive Mechanism for Transmission Expansion in Electricity Networks Theory, Modeling and Application, Revise and Resubmit, The Energy Journal. Hogan, W. (2002a), Financial Transmission Right Incentives: Applications Beyond Hedging Presentation to HEPG Twenty Eighth Plenary Session, May 31, Hogan, W. (2002b), Financial Transmission Right Formulations Mimeo, JFK School of Government, Harvard Electricity Policy Group, Harvard University, Bushnell, J. B., and S. E. Stoft (1997), Improving Private Incentives for Electric Grid Investment Resource and Energy Economics, 19,